In the single field inflation models for which the primordial blackhole production and the secondary gravitational waves are studied, the gravity sector is usually assumed to be the Einstein gravity.
The Einstein gravity however is by no means a complete theory.
From the effective field theory viewpoint, for example, higher curvature terms are expected to arise [1]}.
One such higher curvature term is the Gauss-Bonnet term \(R_{\rm GB}^2 \equiv R^2 - 4R_{\mu \nu }R^{\mu \nu }+R_{\mu \nu \rho \sigma }R^{\mu \nu \rho \sigma }\)  which leads to a relatively well-behaved theory of higher curvature gravity.
Phenomenological aspects of the Gauss-Bonnet correction have been studied by many authors, including dark energy [2]}, [3]}, [4]}, [5]}, [6]}, inflation [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, and gravitational-wave leptogenesis [23]}.
In Ref. [24]}, we investigated a model in which a scalar field \(\varphi \)  is coupled to the Gauss-Bonnet term and discussed the features of a de Sitter-like fixed point as an alternative to cosmic inflation;
in the presence of the Gauss-Bonnet coupling term there may exist a nontrivial de Sitter-like fixed point where the scalar potential term is balanced with the higher curvature Gauss-Bonnet term.
Near the nontrivial fixed point, the standard slow-roll approximation is invalid and the ultra-slow-roll regime of inflation naturally arises.
Furthermore, we pointed out that the primordial curvature power spectrum may become enhanced near the nontrivial de Sitter-like fixed point, which potentially leads to production of primordial blackholes.
In this paper, we investigate the production of primordial blackholes and the scalar-induced second-order gravitational waves in such a setup.
